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Section 1.3 (Part 1)Matrices
Dr. Abdulla Eid
College of Science
MATHS 211: Linear Algebra
Dr. Abdulla Eid (University of Bahrain) Matrices 1 / 12
Goal
We want to learn
1 What a matrix is?
2 How to add or subtract two matrices?
3 How to multiple two matrices?
4 How to find the multiplicative inverse?
5 What is the determinant of a matrix and why it is useful?
6 How to solve system of linear equations using matrices?
Dr. Abdulla Eid (University of Bahrain) Matrices 2 / 12
Goal
We want to learn
1 What a matrix is?
2 How to add or subtract two matrices?
3 How to multiple two matrices?
4 How to find the multiplicative inverse?
5 What is the determinant of a matrix and why it is useful?
6 How to solve system of linear equations using matrices?
Dr. Abdulla Eid (University of Bahrain) Matrices 2 / 12
Goal
We want to learn
1 What a matrix is?
2 How to add or subtract two matrices?
3 How to multiple two matrices?
4 How to find the multiplicative inverse?
5 What is the determinant of a matrix and why it is useful?
6 How to solve system of linear equations using matrices?
Dr. Abdulla Eid (University of Bahrain) Matrices 2 / 12
Goal
We want to learn
1 What a matrix is?
2 How to add or subtract two matrices?
3 How to multiple two matrices?
4 How to find the multiplicative inverse?
5 What is the determinant of a matrix and why it is useful?
6 How to solve system of linear equations using matrices?
Dr. Abdulla Eid (University of Bahrain) Matrices 2 / 12
Goal
We want to learn
1 What a matrix is?
2 How to add or subtract two matrices?
3 How to multiple two matrices?
4 How to find the multiplicative inverse?
5 What is the determinant of a matrix and why it is useful?
6 How to solve system of linear equations using matrices?
Dr. Abdulla Eid (University of Bahrain) Matrices 2 / 12
1- Matrices
Definition 1
A matrix is just a rectangular array of entries. It is described by the rowsand columns.
note: The work matrix is singular. The plural of matrix is matrices(pronounced as‘may tri sees‘).
Example 2
A =
(2 03 2
)B =
(5 6 17 1 2
)C =
(1 2 3
)Usually the matrices are written in the form(
B11 B12 B13
B21 B22 B23
), with B i︸︷︷︸
row
j︸︷︷︸column
Dr. Abdulla Eid (University of Bahrain) Matrices 3 / 12
1- Matrices
Definition 1
A matrix is just a rectangular array of entries. It is described by the rowsand columns.
note: The work matrix is singular. The plural of matrix is matrices(pronounced as‘may tri sees‘).
Example 2
A =
(2 03 2
)B =
(5 6 17 1 2
)C =
(1 2 3
)Usually the matrices are written in the form(
B11 B12 B13
B21 B22 B23
), with B i︸︷︷︸
row
j︸︷︷︸column
Dr. Abdulla Eid (University of Bahrain) Matrices 3 / 12
1- Matrices
Definition 1
A matrix is just a rectangular array of entries. It is described by the rowsand columns.
note: The work matrix is singular. The plural of matrix is matrices(pronounced as‘may tri sees‘).
Example 2
A =
(2 03 2
)
B =
(5 6 17 1 2
)C =
(1 2 3
)Usually the matrices are written in the form(
B11 B12 B13
B21 B22 B23
), with B i︸︷︷︸
row
j︸︷︷︸column
Dr. Abdulla Eid (University of Bahrain) Matrices 3 / 12
1- Matrices
Definition 1
A matrix is just a rectangular array of entries. It is described by the rowsand columns.
note: The work matrix is singular. The plural of matrix is matrices(pronounced as‘may tri sees‘).
Example 2
A =
(2 03 2
)B =
(5 6 17 1 2
)
C =(1 2 3
)Usually the matrices are written in the form(
B11 B12 B13
B21 B22 B23
), with B i︸︷︷︸
row
j︸︷︷︸column
Dr. Abdulla Eid (University of Bahrain) Matrices 3 / 12
1- Matrices
Definition 1
A matrix is just a rectangular array of entries. It is described by the rowsand columns.
note: The work matrix is singular. The plural of matrix is matrices(pronounced as‘may tri sees‘).
Example 2
A =
(2 03 2
)B =
(5 6 17 1 2
)C =
(1 2 3
)
Usually the matrices are written in the form(B11 B12 B13
B21 B22 B23
), with B i︸︷︷︸
row
j︸︷︷︸column
Dr. Abdulla Eid (University of Bahrain) Matrices 3 / 12
1- Matrices
Definition 1
A matrix is just a rectangular array of entries. It is described by the rowsand columns.
note: The work matrix is singular. The plural of matrix is matrices(pronounced as‘may tri sees‘).
Example 2
A =
(2 03 2
)B =
(5 6 17 1 2
)C =
(1 2 3
)
Usually the matrices are written in the form(B11 B12 B13
B21 B22 B23
), with B i︸︷︷︸
row
j︸︷︷︸column
Dr. Abdulla Eid (University of Bahrain) Matrices 3 / 12
1- Matrices
Definition 1
A matrix is just a rectangular array of entries. It is described by the rowsand columns.
note: The work matrix is singular. The plural of matrix is matrices(pronounced as‘may tri sees‘).
Example 2
A =
(2 03 2
)B =
(5 6 17 1 2
)C =
(1 2 3
)Usually the matrices are written in the form(
B11 B12 B13
B21 B22 B23
), with B i︸︷︷︸
row
j︸︷︷︸column
Dr. Abdulla Eid (University of Bahrain) Matrices 3 / 12
Definition 3
An m× n–matrix is a rectangular array consists of m rows and n columns.
A =
A11 A12 · · · A1n
A21 A22 · · · A2n
· · · ·· · · ·· · · ·
An1 An2 · · · Ann
= (Aij )m×n
where Aij is the entry in the row i and column j .
Dr. Abdulla Eid (University of Bahrain) Matrices 4 / 12
Example 4
Let
A =
3 −2 7 32 1 −1 −54 3 2 10 8 0 2
1 What is the size of A?
2 Find A21,A42,A32,A34,A44,A55.
3 What are the entries of the second row?
Dr. Abdulla Eid (University of Bahrain) Matrices 5 / 12
Definition 5
If A is a matrix, the transpose of A is a new matrix AT formed byinterchanging the rows and the columns of A, i.e.,
AT = (Aji )
Example 6
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
AT =
(6 2−3 4
)and (AT )T =
(6 −32 4
)
Dr. Abdulla Eid (University of Bahrain) Matrices 6 / 12
Definition 5
If A is a matrix, the transpose of A is a new matrix AT formed byinterchanging the rows and the columns of A, i.e.,
AT = (Aji )
Example 6
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
AT =
(6 2−3 4
)and (AT )T =
(6 −32 4
)
Dr. Abdulla Eid (University of Bahrain) Matrices 6 / 12
Definition 5
If A is a matrix, the transpose of A is a new matrix AT formed byinterchanging the rows and the columns of A, i.e.,
AT = (Aji )
Example 6
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
AT =
(6 2−3 4
)and (AT )T =
(6 −32 4
)
Dr. Abdulla Eid (University of Bahrain) Matrices 6 / 12
Definition 5
If A is a matrix, the transpose of A is a new matrix AT formed byinterchanging the rows and the columns of A, i.e.,
AT = (Aji )
Example 6
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
AT =
(6 2−3 4
)and (AT )T =
(6 −32 4
)
Dr. Abdulla Eid (University of Bahrain) Matrices 6 / 12
Definition 5
If A is a matrix, the transpose of A is a new matrix AT formed byinterchanging the rows and the columns of A, i.e.,
AT = (Aji )
Example 6
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
AT =
(6 2−3 4
)and (AT )T =
(6 −32 4
)Dr. Abdulla Eid (University of Bahrain) Matrices 6 / 12
Example 7
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
BT =
2 71 13 6
and (BT )T =
(5 6 17 1 2
)
2
CT =
3125
and (CT )T =(3 1 2 5
)
Dr. Abdulla Eid (University of Bahrain) Matrices 7 / 12
Example 7
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
BT =
2 71 13 6
and (BT )T =
(5 6 17 1 2
)
2
CT =
3125
and (CT )T =(3 1 2 5
)
Dr. Abdulla Eid (University of Bahrain) Matrices 7 / 12
Example 7
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
BT =
2 71 13 6
and (BT )T =
(5 6 17 1 2
)
2
CT =
3125
and (CT )T =(3 1 2 5
)
Dr. Abdulla Eid (University of Bahrain) Matrices 7 / 12
Example 7
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
BT =
2 71 13 6
and (BT )T =
(5 6 17 1 2
)
2
CT =
3125
and (CT )T =(3 1 2 5
)
Dr. Abdulla Eid (University of Bahrain) Matrices 7 / 12
Example 7
Find the transpose MT and (MT )T .
A =
(6 −32 4
)B =
(2 1 37 1 6
)C =
(3 1 2 5
)
Solution:
1
BT =
2 71 13 6
and (BT )T =
(5 6 17 1 2
)
2
CT =
3125
and (CT )T =(3 1 2 5
)
Dr. Abdulla Eid (University of Bahrain) Matrices 7 / 12
Note:1 (AT )T = A.
2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→ y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if
AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→ y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→ y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→ y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→ y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→ y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3,
2y = −8→ y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→
y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→ y = −4, 3z = 0→
z = 0
Solution Set = {(−3,−4, 0)}
Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Note:1 (AT )T = A.2 A matrix A is called symmetric if AT = A.
Question: When two matrices are equal?
Definition 8
Two matrices A and B are equal if they have the same size and the sameentries at the same position, i.e.,
Aij = Bij
Example 9
Solve the matrix equation4 2 1x 2y 3z0 1 2
=
4 2 1−3 −8 00 1 2
Solution:
x = −3, 2y = −8→ y = −4, 3z = 0→ z = 0
Solution Set = {(−3,−4, 0)}Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 12
Special Matrices
Zero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n
= (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n
“zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if
m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if
it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special MatricesZero matrix 0m×n = (0)m×n “zero everywhere“.
(0 0
),
(0 0 00 0 0
),
0 0 00 0 00 0 0
,
(00
)
Square matrix if m = n (having the same number of rows andcolumns).
(3)
,
(3 21 −5
),
6 5 −11 3 38 −9 0
Diagonal matrix if it is a square matrix (m = n) and all entries off themain diagonal are zeros.
(3 00 −5
),
6 0 00 3 00 0 6
,
3 0 0 00 −5 0 00 0 −11 00 0 0 2
Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 12
Special Matrices
Upper Diagonal matrix if
it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if
it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Upper Diagonal matrix if it has zeros below the main diagonal(entries are ‘upper‘ the main diagonal).
(3 50 −5
),
6 3 50 3 −20 0 6
,
3 1 2 −70 −5 −4 60 0 −11 60 0 0 2
Lower Diagonal matrix if it has zeros above the main diagonal(entries are ‘lower‘ the main diagonal).
(3 07 −5
),
6 0 03 3 04 7 6
,
3 0 0 06 −5 0 0−8 −4 −11 01 4 7 2
Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 12
Special Matrices
Row vector is
a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)
Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is
a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if
m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
,
I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
),
I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
,
I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Special Matrices
Row vector is a matirx with only one row.(2 3
),
(5 13 12
),
(7 3 0 −2 6
),
(0 0 0
)Column vector is a matrix with only one column.
(36
),
31−5
,
6180
Identity matrix In if m = n and has one in the main diagonal and zeroelsewhere.
I1 =(1)
, I2 =
(1 00 1
), I3 =
1 0 00 1 00 0 1
, I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 12
Definition 10
If A is a square matrix, the trace of A is sum of the main diagonal
tr(A) =n
∑i=1
Aii
Example 11
Find the trace of the following matrices.
A =
(6 −32 4
)B =
2 1 37 1 60 0 −3
C =
3 1 2 5
13 0 22 1533 41 13 6534 15 2 −10
Solution:
Dr. Abdulla Eid (University of Bahrain) Matrices 12 / 12
Definition 10
If A is a square matrix, the trace of A is sum of the main diagonal
tr(A) =n
∑i=1
Aii
Example 11
Find the trace of the following matrices.
A =
(6 −32 4
)B =
2 1 37 1 60 0 −3
C =
3 1 2 5
13 0 22 1533 41 13 6534 15 2 −10
Solution:
Dr. Abdulla Eid (University of Bahrain) Matrices 12 / 12
Definition 10
If A is a square matrix, the trace of A is sum of the main diagonal
tr(A) =n
∑i=1
Aii
Example 11
Find the trace of the following matrices.
A =
(6 −32 4
)B =
2 1 37 1 60 0 −3
C =
3 1 2 5
13 0 22 1533 41 13 6534 15 2 −10
Solution:
Dr. Abdulla Eid (University of Bahrain) Matrices 12 / 12
Definition 10
If A is a square matrix, the trace of A is sum of the main diagonal
tr(A) =n
∑i=1
Aii
Example 11
Find the trace of the following matrices.
A =
(6 −32 4
)B =
2 1 37 1 60 0 −3
C =
3 1 2 5
13 0 22 1533 41 13 6534 15 2 −10
Solution:
Dr. Abdulla Eid (University of Bahrain) Matrices 12 / 12
